We consider the Cauchy problem for an $n\times n$ strictly hyperbolic system of balance laws $u_t+f(u)_x=g(x,u)$, $x\in\mathbb{R}$, $t>0$, $\|g(x,\cdot)\|_{\mathbf{C}^2}\leq\tilde{M}(x)\in\mathbf{L}^1$, endowed with the initial data $u(0,.)=u_o\in\mathbf{L}^1\cap\mathbf{BV}(\mathbb{R};\mathbb{R}^n)$. Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., $|\lambda_i(u)|\geq c>0$ for all $i\in\{1,\dots,n\}$. Assuming that the $\mathbf{L}^1$ norms of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{\mathbf{BV}(\mathbb{R})}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D.~Amadori, L.~Gosse, and G.~Guerra, {\it Arch.~Ration.~Mech.~Anal.}, 162 (2002), pp.~327--366] to unbounded (in $\mathbf{L}^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.

Guerra, G., Marcellini, F., Schleper, V. (2009). Balance laws with integrable unbounded sources. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 41(3), 1164-1189 [10.1137/080735436].

Balance laws with integrable unbounded sources

GUERRA, GRAZIANO;MARCELLINI, FRANCESCA;
2009

Abstract

We consider the Cauchy problem for an $n\times n$ strictly hyperbolic system of balance laws $u_t+f(u)_x=g(x,u)$, $x\in\mathbb{R}$, $t>0$, $\|g(x,\cdot)\|_{\mathbf{C}^2}\leq\tilde{M}(x)\in\mathbf{L}^1$, endowed with the initial data $u(0,.)=u_o\in\mathbf{L}^1\cap\mathbf{BV}(\mathbb{R};\mathbb{R}^n)$. Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., $|\lambda_i(u)|\geq c>0$ for all $i\in\{1,\dots,n\}$. Assuming that the $\mathbf{L}^1$ norms of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{\mathbf{BV}(\mathbb{R})}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D.~Amadori, L.~Gosse, and G.~Guerra, {\it Arch.~Ration.~Mech.~Anal.}, 162 (2002), pp.~327--366] to unbounded (in $\mathbf{L}^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.
Articolo in rivista - Articolo scientifico
hyperbolic balance laws, unbounded sources, pipes with discontinuous cross sections
English
2009
41
3
1164
1189
open
Guerra, G., Marcellini, F., Schleper, V. (2009). Balance laws with integrable unbounded sources. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 41(3), 1164-1189 [10.1137/080735436].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/7386
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